IIT JEE 2008 Paper II

1. A particle P starts from the point z0 = 1 + 2i, where i = √–1. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z1. From z1 the particle moves √2 units in the direction of the vector i^+ j^and then it moves through an angle π /2 in anticlockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by

&nbsp 6 + 7i

&nbsp – 7 + 6i

&nbsp 7 + 6i

&nbsp – 6 + 7i

Hint

Half-n-half Clue

2.

2.

Let the function g : (− ∞, ∞) →

(−

π

2

,

π

2

) be given by g(u) = 2 tan−1 (eu) −

π

2

. Then, g is

&nbsp even and is strictly increasing in (0, ∞)

&nbsp odd and is strictly decreasing in (– ∞, ∞)

&nbsp odd and is strictly increasing in (– ∞, ∞)

&nbsp neither even nor odd, but is strictly increasing in (– ∞, ∞)

Half-n-half Clue

3.

3. Consider a branch of the hyperbola

x2 – 2 y2 – 2 √2 x – 4 √2 y – 6 = 0

with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

6. An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

&nbsp 2, 4 or 8

&nbsp 3, 6 or 9

&nbsp 4 or 8

&nbsp 5 or 10

Half-n-half Clue

7.

7. Let two non-collinear unit vectors a^and b^ form an acute angle. A point P moves so that at any time t the position vector OP→ (where O is the origin) is given by a^cos t + b^sin t. When P is farthest from origin O, let M be the length of OP→ be the unit vector along OP→. Then